STAT 511
Homework 1
Due Date: 11:00 A.M., Wednesday, January 18
Please review Ken Koehler’s notes on working with vectors and matrices available at
http://www.public.iastate.edu/∼kkoehler/stat511/sect2.4page.pdf
Here are a few points to note when examining the notes.
• We will use the notation A0 to denote the transpose of the matrix A rather than AT as in Dr. Koehler’s
notes.
• We will use R rather than S-PLUS as in Dr. Koehler’s notes. However, Dr. Koehler’s S-PLUS code
should work fine in R.
• On slide 115, “frac13” should be 13 .
Once you have reviewed the notes, please answer the following questions. It is possible and preferable
to complete the problems without the use of a computer. You are encouraged to check your work with the
help of R.
1. Determine the rank of the following matrix.

3 2 7 1
5
 −1 3 5 1
3 
4 2 8 7 −6

2. Consider the design matrix on slide 19 of the lecture notes available at
http://www.public.iastate.edu/∼dnett/S511/01Introduction.pdf
What is the largest possible rank of this matrix, and what must be true about x1 , . . . , x8 to achieve
that maximum rank?
3. Consider the design matrix on page 30 of the lecture notes available at
http://www.public.iastate.edu/∼dnett/S511/01Introduction.pdf
(a) Determine the rank of design matrix.
(b) The column space of a matrix is the set of vectors spanned by the columns of the matrix. Give
another matrix with the same column space as the design matrix.
4. Consider the subset of IR3 given by S = {x ∈ IR3 : 10 x = 0}. (Here 1 = [1, 1, 1]0 .)
(a) Show that S is a vector space.
(b) Provide a basis for S.
(c) State the dimension of S.
5. If x and y are random vectors, Cov(x, y) is defined as E[{x − E(x)}{y − E(y)}0 ]. (Note that it is
not necessary for x and y to be of the same order.) Suppose A and B are matrices of constants, a and
b are vectors of constants, and z is a random vector with mean µ and variance Σ. Give a simplified
expression for Cov(Az + a, Bz + b).
1
6. Suppose A is a matrix. Prove that A = 0 if and only if tr(A0 A) = 0.
7. Suppose a1 , a2 , a3 , a4 ∈ IR. Consider the matrix


a1 0 0 0
 0 a2 0 0 

diag(a1 , a2 , a3 , a4 ) ≡ 
 0 0 a3 0  .
0 0 0 a4
(a) Find the determinant of diag(a1 , a2 , a3 , a4 ).
(b) Find the eigenvalues of diag(a1 , a2 , a3 , a4 ).
8. Result 1.10 (iii) is known as the cyclic property of the trace. Use this property to prove Result 1.10
(iv).
9. Prove Results 1.11 (iv), (vi), and (vii). You may use any results stated before slide 86 of Dr. Koehler’s
notes. You are not allowed to use results stated after slide 86 without first proving those results.
10. Result 1.12 is sometimes known as the Spectral Decomposition Theorem. This result is very useful
and powerful. Use this result along with the cyclic property of the trace to prove Result 1.11 (v).
11. Find a so that
   

a 
6
 4
 1  is in the span of  1  ,  0  .


8
0
−2

12. Suppose W1 and W2 are random variables. Suppose Var(W1 ) = 4, Var(W2 ) = 2, and Cov(W1 , W2 ) =
−1. Find
W1 + W2
Var
.
W1 − W2
2